Adiabatic approximation of the SchrödingerPoisson system with a partial confinement

Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the three-dimensional Schrodinger-Poisson system. The small parameter " is the scaled width of the electron gas and appears as the lengthscale on which a one dimen-sional conning potential varies. The rigorous " ! 0 limit is performed by pro-jecting the three dimensional wavefunction on the eigenfunctions correspond-ing to the conning potential. This leads to a two-dimensional Schrodinger-Poisson system with a modied Poisson equation keeping track of the third dimension. This limit model is proven to be a rst-order approximation of the initial model. An intermediate model, called the \2.5D adiabatic model " is then introduced. It shares the same structure as the limit model but is shown to be a second-order approximation of the 3D model